Thoroughly cheated and victimized. Or else enjoying the perfect shore leave.
“Stewed” is sometimes added for further effect. (Dictionary of Slang and Unconventional English by Partridge) Adding “and sold down the river” also means “very drunk.”
Since the phrase could mean being completely put through some process, it may originally have referred to the process of becoming a full-fledged sailor. In the Pacific Fleet in WWII, if you were in the Navy, the phrase meant that you had landed in a foreign port and had done everything of consequence in that port. First, you had been screwed (serviced by the local prostitutes). Second, you’d been blued (had a local tailor make you a couple of sets of Dress Blue Uniforms), and third, you were tattooed, which doesn’t require explanation since tattooing has always been associated with sailors. The connection to a sailor on a wild shore leave is traditional.
Another theory, much harsher, follows the current definition of “screwed” as being cheated. “Blued” may come from the earlier “blewed,” meaning “robbed.” And “tattooed” is recorded in Chapman’s Dictionary of American Slang as “struck rapidly and repeatedly.” So, in this sense, you’ve been cheated, beaten up, and robbed.
A similar phrase, the British “Bitched, buggered, and bewildered” (and “far from home”), is a parody of the popular song from the 1940 musical hit Pal Joey: “Bewitched, Bothered, and Bewildered.”
Sleeze Beez, a glam metal band formed in 1987 in the Netherlands, released their 1989 album Screwed Blued & Tattooed, which peaked at number 115 on the Billboard 200 album chart.
I like the Navy connection, but I’d certainly think twice about joining up.